The physical picture of BPS states as multi-centered black holes suggests one way to achieve this goal, namely to decompose stable BPS states of charge \(\gamma \)  into bound states of
elementary constituents of charge \(\gamma _i\) , with a hierarchical structure determined by
attractor flow trees [1]}, [2]}. As we review in more detail
in §REF , the latter are rooted trees decorated
with charges \(\gamma _e\)  along the edges,
embedded in \(\Pi \subset \operatorname{Stab}(\mathfrak {Y})\)  such that the root vertex is mapped to the desired point \(z\in \Pi \) ,
edges follow the gradient flow (also known as attractor flow [3]}) for the modulus of the central charge \(|Z(\gamma _e)|^2\)  along the slice \(\Pi \) , and split at vertices on
walls of marginal stability where the central charges of the incoming and descending charges
become aligned. The aforementioned constituents \(\gamma _i\)  arise as the end points (or leaves)
of the tree, where the central charge is attracted to a local minimum of \(|Z(\gamma _i)|\)  along \(\Pi \) ,
or to a conifold point where \(Z(\gamma _i)=0\) . We denote by \(Z_{\gamma _i}(\gamma )\)  the central charge at this local minimum, and by
\(\Omega _\star (\gamma _i)\Omega _{Z_{\gamma _i}}(\gamma _i)\)  the corresponding value of
the DT invariant, known as attractor index.
